Pub Date : 2017-12-05DOI: 10.4310/ARKIV.2019.V57.N1.A5
L. Guerrieri
In this paper we study the Weak Lefschetz property of two classes of standard graded Artinian Gorenstein algebras associated in a natural way to the Ap'ery set of numerical semigroups. To this aim we also prove a general result about the transfer of Weak Lefschetz property from an Artinian Gorenstein algebra to its quotients modulo a colon ideal.
{"title":"Lefschetz properties of Gorenstein graded algebras associated to the Apéry set of a numerical semigroup","authors":"L. Guerrieri","doi":"10.4310/ARKIV.2019.V57.N1.A5","DOIUrl":"https://doi.org/10.4310/ARKIV.2019.V57.N1.A5","url":null,"abstract":"In this paper we study the Weak Lefschetz property of two classes of standard graded Artinian Gorenstein algebras associated in a natural way to the Ap'ery set of numerical semigroups. To this aim we also prove a general result about the transfer of Weak Lefschetz property from an Artinian Gorenstein algebra to its quotients modulo a colon ideal.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2017-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47312899","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-11-29DOI: 10.4310/ARKIV.2019.v57.n1.a11
Alessio Sammartano
Let S be a numerical semigroup, R the local ring of the monomial curve singularity associated to S, and G its associated graded ring. In this paper we provide a sharp upper bound for the least positive integer in S in terms of the codimension and the maximum degree of the equations of G, when G is not a complete intersection. A special case of this result settles a question of J. Herzog and D. Stamate.
{"title":"On the multiplicity of tangent cones of monomial curves","authors":"Alessio Sammartano","doi":"10.4310/ARKIV.2019.v57.n1.a11","DOIUrl":"https://doi.org/10.4310/ARKIV.2019.v57.n1.a11","url":null,"abstract":"Let S be a numerical semigroup, R the local ring of the monomial curve singularity associated to S, and G its associated graded ring. In this paper we provide a sharp upper bound for the least positive integer in S in terms of the codimension and the maximum degree of the equations of G, when G is not a complete intersection. A special case of this result settles a question of J. Herzog and D. Stamate.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2017-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49282704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-11-15DOI: 10.4310/arkiv.2019.v57.n2.a5
F. Goncharov, R. Novikov
We consider weighted ray-transforms $P_W$ (weighted Radon transforms along straight lines) in $R^d, ,d geq 2$, with strictly positive weights $W$. We construct an example of such a transform with non-trivial kernel in the space of infinitely smooth compactly supported functions on $R^d$. In addition, the constructed weight W is rotation-invariant continuous and is infinitely smooth almost everywhere on $R^d times S^{d-1}$. In particular, by this construction we give counterexamples to some well-known injectivity results for weighted ray transforms for the case when the regularity of $W$ is slightly relaxed.
我们考虑$R^d, ,d geq 2$中的加权射线变换$P_W$(沿直线的加权Radon变换)，具有严格的正权$W$。我们在$R^d$上的无限光滑紧支持函数空间中构造了一个具有非平凡核的这种变换的例子。此外，所构造的权值W在$R^d times S^{d-1}$上几乎处处是无限光滑的，并且是旋转不变连续的。特别地，通过这种构造，我们给出了对于$W$的正则性稍松弛的情况下加权射线变换的一些著名的注入性结果的反例。
{"title":"A breakdown of injectivity for weighted ray transforms in multidimensions","authors":"F. Goncharov, R. Novikov","doi":"10.4310/arkiv.2019.v57.n2.a5","DOIUrl":"https://doi.org/10.4310/arkiv.2019.v57.n2.a5","url":null,"abstract":"We consider weighted ray-transforms $P_W$ (weighted Radon transforms along straight lines) in $R^d, ,d geq 2$, with strictly positive weights $W$. We construct an example of such a transform with non-trivial kernel in the space of infinitely smooth compactly supported functions on $R^d$. In addition, the constructed weight W is rotation-invariant continuous and is infinitely smooth almost everywhere on $R^d times S^{d-1}$. In particular, by this construction we give counterexamples to some well-known injectivity results for weighted ray transforms for the case when the regularity of $W$ is slightly relaxed.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2017-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393689","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-10-10DOI: 10.4310/arkiv.2020.v58.n1.a7
P. Hazard
We give examples of endomorphisms in dimension one with infinite topological entropy which are $alpha$-H"older and $(1,p)$-Sobolev for all $0leqalpha<1$ and $1leq p
我们给出了具有无限拓扑熵的一维自同态的例子，它们是$alpha$ -Hölder和$(1,p)$ -Sobolev对于所有$0leqalpha<1$和$1leq p
{"title":"Maps in dimension one with infinite entropy","authors":"P. Hazard","doi":"10.4310/arkiv.2020.v58.n1.a7","DOIUrl":"https://doi.org/10.4310/arkiv.2020.v58.n1.a7","url":null,"abstract":"We give examples of endomorphisms in dimension one with infinite topological entropy which are $alpha$-H\"older and $(1,p)$-Sobolev for all $0leqalpha<1$ and $1leq p<infty$. This is constructed within a family of endomorphisms with infinite topological entropy and which traverse all $alpha$-H\"older and $(1,p)$-Sobolev classes. Finally, we also give examples of endomorphisms, also in dimension one, which lie in the big and little Zygmund classes, answering a question of M. Benedicks.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2017-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-10-04DOI: 10.4310/ARKIV.2019.V57.N1.A6
Christian Hagg
We give an explicit formula for the logarithmic potential of the asymptotic zero-counting measure of the sequence $left{frac{mathrm{d}^n}{mathrm{d}z^n}left(R(z)exp{T(z)}right)right}$. Here, $R(z)$ is a rational function with at least two poles, all of which are distinct, and $T(z)$ is a polynomial. This is an extension of a recent measure-theoretic refinement of Polya's Shire theorem for rational functions.
{"title":"The asymptotic zero-counting measure of iterated derivaties of a class of meromorphic functions","authors":"Christian Hagg","doi":"10.4310/ARKIV.2019.V57.N1.A6","DOIUrl":"https://doi.org/10.4310/ARKIV.2019.V57.N1.A6","url":null,"abstract":"We give an explicit formula for the logarithmic potential of the asymptotic zero-counting measure of the sequence $left{frac{mathrm{d}^n}{mathrm{d}z^n}left(R(z)exp{T(z)}right)right}$. Here, $R(z)$ is a rational function with at least two poles, all of which are distinct, and $T(z)$ is a polynomial. This is an extension of a recent measure-theoretic refinement of Polya's Shire theorem for rational functions.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2017-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393574","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-09-01DOI: 10.4310/ARKIV.2017.V55.N1.A7
J. Jo
We show that if $X$ is a cocompact $Gtextrm{-}CW$-complex such that each isotropy subgroup $G_sigma$ is $L^{(2)}$-good over an arbitrary commutative ring $k$, then $X$ satisfies some fixed-point formula which is an $L^{(2)}$-analogue of Brown’s formula in 1982. Using this result we present a fixed point formula for a cocompact proper $Gtextrm{-}CW$-complex which relates the equivariant $L^{(2)}$-Euler characteristic of a fixed point $CW$-complex $X^s$ and the Euler characteristic of $X/G$. As corollaries, we prove Atiyah’s theorem in 1976, Akita’s formula in 1999 and a result of Chatterji–Mislin in 2009. We also show that if X is a free $Gtextrm{-}CW$-complex such that $C_{*} (X)$ is chain homotopy equivalent to a chain complex of finitely generated projective $Z pi_1 (X)$-modules of finite length and $X$ satisfies some fixed-point formula over $mathbb{Q}$ or $mathbb{C}$ which is an $L^{(2)}$-analogue of Brown’s formula, then $chi (X/G) = chi^{(2)} (X)$. As an application, we prove that the weak Bass conjecture holds for any finitely presented group $G$ satisfying the following condition: for any finitely dominated $CW$-complex $Y$ with $pi_1 (Y)=G, widetilde{Y}$ satisfies some fixed-point formula over $mathbb{Q}$ or $mathbb{C}$ which is an $L^{(2)}$-analogue of Brown’s formula.
{"title":"Equivariant $L^2$-Euler characteristics of $Gtextrm{-}CW$-complexes","authors":"J. Jo","doi":"10.4310/ARKIV.2017.V55.N1.A7","DOIUrl":"https://doi.org/10.4310/ARKIV.2017.V55.N1.A7","url":null,"abstract":"We show that if $X$ is a cocompact $Gtextrm{-}CW$-complex such that each isotropy subgroup $G_sigma$ is $L^{(2)}$-good over an arbitrary commutative ring $k$, then $X$ satisfies some fixed-point formula which is an $L^{(2)}$-analogue of Brown’s formula in 1982. Using this result we present a fixed point formula for a cocompact proper $Gtextrm{-}CW$-complex which relates the equivariant $L^{(2)}$-Euler characteristic of a fixed point $CW$-complex $X^s$ and the Euler characteristic of $X/G$. As corollaries, we prove Atiyah’s theorem in 1976, Akita’s formula in 1999 and a result of Chatterji–Mislin in 2009. We also show that if X is a free $Gtextrm{-}CW$-complex such that $C_{*} (X)$ is chain homotopy equivalent to a chain complex of finitely generated projective $Z pi_1 (X)$-modules of finite length and $X$ satisfies some fixed-point formula over $mathbb{Q}$ or $mathbb{C}$ which is an $L^{(2)}$-analogue of Brown’s formula, then $chi (X/G) = chi^{(2)} (X)$. As an application, we prove that the weak Bass conjecture holds for any finitely presented group $G$ satisfying the following condition: for any finitely dominated $CW$-complex $Y$ with $pi_1 (Y)=G, widetilde{Y}$ satisfies some fixed-point formula over $mathbb{Q}$ or $mathbb{C}$ which is an $L^{(2)}$-analogue of Brown’s formula.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":"55 1","pages":"155-164"},"PeriodicalIF":0.7,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70392975","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-09-01DOI: 10.4310/ARKIV.2017.V55.N1.A3
Kaneko Hajime
Many mathematicians have studied the algebraic independence over Q of the values of gap series, and the values of lacunary series satisfying functional equations of Mahler type. In this paper, we give a new criterion for the algebraic independence over Q of the values ∑∞ n=0 t(n)β −n for distinct sequences (t(n))n=0 of nonnegative integers, where β is a fixed Pisot or Salem number. Our criterion is applicable to certain power series which are not lacunary. Moreover, our criterion does not use functional equations. Consequently, we deduce the algebraic independence of certain values ∑∞ n=0 t1(n)β −n, . . . , ∑∞ n=0 tr(n)β −n satisfying lim n→∞,ti−1(n) ̸=0 ti(n) ti−1(n) = ∞ (i = 2, . . . , r) for any positive real number M . 1 The transcendence of the values of power series with bounded coefficients We introduce notation which we use throughout this paper. Let N (resp. Z) be the set of nonnegative integers (resp. positive integers). For a real number x, we denote the integral and fractional parts of x by ⌊x⌋ and {x}, respectively. We use the Landau symbols o,O, and the Vinogradov symbols ≫,≪ with their regular meanings. For a sequence of integers t = (tn) ∞ n=0, put S(t) := {n ∈ N | tn ̸= 0}. and
{"title":"Algebraic independence of the values of power series with unbounded coefficients","authors":"Kaneko Hajime","doi":"10.4310/ARKIV.2017.V55.N1.A3","DOIUrl":"https://doi.org/10.4310/ARKIV.2017.V55.N1.A3","url":null,"abstract":"Many mathematicians have studied the algebraic independence over Q of the values of gap series, and the values of lacunary series satisfying functional equations of Mahler type. In this paper, we give a new criterion for the algebraic independence over Q of the values ∑∞ n=0 t(n)β −n for distinct sequences (t(n))n=0 of nonnegative integers, where β is a fixed Pisot or Salem number. Our criterion is applicable to certain power series which are not lacunary. Moreover, our criterion does not use functional equations. Consequently, we deduce the algebraic independence of certain values ∑∞ n=0 t1(n)β −n, . . . , ∑∞ n=0 tr(n)β −n satisfying lim n→∞,ti−1(n) ̸=0 ti(n) ti−1(n) = ∞ (i = 2, . . . , r) for any positive real number M . 1 The transcendence of the values of power series with bounded coefficients We introduce notation which we use throughout this paper. Let N (resp. Z) be the set of nonnegative integers (resp. positive integers). For a real number x, we denote the integral and fractional parts of x by ⌊x⌋ and {x}, respectively. We use the Landau symbols o,O, and the Vinogradov symbols ≫,≪ with their regular meanings. For a sequence of integers t = (tn) ∞ n=0, put S(t) := {n ∈ N | tn ̸= 0}. and","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":"55 1","pages":"61-87"},"PeriodicalIF":0.7,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393305","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-09-01DOI: 10.4310/ARKIV.2017.V55.N1.A5
V. Exnerov'a, J. Malý, O. Martio
Moduli of path families are widely used to mark curves which may be neglected for some applications. We introduce ordinary and approximation modulus with respect to Banach function spaces. While these moduli lead to the same result in reflexive spaces, we show that there are important path families (like paths tangent to a given set) which can be labeled as negligible by the approximation modulus with respect to the Lorentz Lp,1-space for an appropriate p, in particular, to the ordinary L1-space if p=1, but not by the ordinary modulus with respect to the same space.
{"title":"Modulus in Banach function spaces","authors":"V. Exnerov'a, J. Malý, O. Martio","doi":"10.4310/ARKIV.2017.V55.N1.A5","DOIUrl":"https://doi.org/10.4310/ARKIV.2017.V55.N1.A5","url":null,"abstract":"Moduli of path families are widely used to mark curves which may be neglected for some applications. We introduce ordinary and approximation modulus with respect to Banach function spaces. While these moduli lead to the same result in reflexive spaces, we show that there are important path families (like paths tangent to a given set) which can be labeled as negligible by the approximation modulus with respect to the Lorentz Lp,1-space for an appropriate p, in particular, to the ordinary L1-space if p=1, but not by the ordinary modulus with respect to the same space.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":"55 1","pages":"105-130"},"PeriodicalIF":0.7,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47287712","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-09-01DOI: 10.4310/ARKIV.2017.V55.N1.A11
M. Montenegro, Adilson E. Presoto
Let F :RN→RN be a locally Lipschitz continuous function. We prove that F is a global homeomorphism or only injective, under suitable assumptions on the subdifferential ∂F (x). We use variational methods, nonsmooth inverse function theorem and extensions of the Hadamard-Levy Theorem. We also address questions on the Markus-Yamabe conjecture.
{"title":"Invertibility of nonsmooth mappings","authors":"M. Montenegro, Adilson E. Presoto","doi":"10.4310/ARKIV.2017.V55.N1.A11","DOIUrl":"https://doi.org/10.4310/ARKIV.2017.V55.N1.A11","url":null,"abstract":"Let F :RN→RN be a locally Lipschitz continuous function. We prove that F is a global homeomorphism or only injective, under suitable assumptions on the subdifferential ∂F (x). We use variational methods, nonsmooth inverse function theorem and extensions of the Hadamard-Levy Theorem. We also address questions on the Markus-Yamabe conjecture.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":"55 1","pages":"217-228"},"PeriodicalIF":0.7,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70392745","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-09-01DOI: 10.4310/ARKIV.2017.V55.N1.A4
Nguyen Thu Hang, T. N. Trung
We prove that the cover ideals of all unimodular hypergraphs have the nonincreasing depth function property. Furthermore, we show that the index of depth stability of these ideals is bounded by the number of variables. Introduction Let R=k[x1, ..., xn] be a polynomial ring over a given field k, and let I be a homogeneous ideal in R. It is known by Brodmann [3] that depth(R/I) takes a constant value for large s. Moreover, lim s→∞ depthR/I dimR− (I), where (I) is the analytic spread of I. The index of depth stability of I is defined by dstab(I) :=min { s0 1 |depthS/I =depthS/I0 for all s s0 } . Two natural questions arise from Brodmann’s theorem: (1) What is the nature of the function s →depthR/Is for s dstab(I)? (2) What is a reasonable bound for dstab(I)? On the nature of the function s →depthR/Is for s 1, which is called the depth function of I, Herzog and Hibi [10] conjectured that the depth function of ideals can be any convergent nonnegative integer valued function. The answer is affirmative for bounded increasing functions (see [10]) and non-increasing functions (see [8]). The behavior of depth functions, even for monomial ideals, is complicated (see e.g. [1]). Squarefree monomial ideals behave considerably better than monomial
{"title":"The behavior of depth functions of cover ideals of unimodular hypergraphs","authors":"Nguyen Thu Hang, T. N. Trung","doi":"10.4310/ARKIV.2017.V55.N1.A4","DOIUrl":"https://doi.org/10.4310/ARKIV.2017.V55.N1.A4","url":null,"abstract":"We prove that the cover ideals of all unimodular hypergraphs have the nonincreasing depth function property. Furthermore, we show that the index of depth stability of these ideals is bounded by the number of variables. Introduction Let R=k[x1, ..., xn] be a polynomial ring over a given field k, and let I be a homogeneous ideal in R. It is known by Brodmann [3] that depth(R/I) takes a constant value for large s. Moreover, lim s→∞ depthR/I dimR− (I), where (I) is the analytic spread of I. The index of depth stability of I is defined by dstab(I) :=min { s0 1 |depthS/I =depthS/I0 for all s s0 } . Two natural questions arise from Brodmann’s theorem: (1) What is the nature of the function s →depthR/Is for s dstab(I)? (2) What is a reasonable bound for dstab(I)? On the nature of the function s →depthR/Is for s 1, which is called the depth function of I, Herzog and Hibi [10] conjectured that the depth function of ideals can be any convergent nonnegative integer valued function. The answer is affirmative for bounded increasing functions (see [10]) and non-increasing functions (see [8]). The behavior of depth functions, even for monomial ideals, is complicated (see e.g. [1]). Squarefree monomial ideals behave considerably better than monomial","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":"55 1","pages":"89-104"},"PeriodicalIF":0.7,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393370","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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